272 lines
11 KiB
Text
272 lines
11 KiB
Text
/-
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Copyright (c) 2015 Microsoft Corporation. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Author: Leonardo de Moura
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The length of a list is encoded into its type.
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It is implemented as a subtype.
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-/
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import logic data.list data.fin
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open nat list subtype function
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definition fixed_list [reducible] (A : Type) (n : nat) := {l : list A | length l = n}
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namespace fixed_list
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variables {A B C : Type}
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theorem induction_on [recursor 4] {P : ∀ {n}, fixed_list A n → Prop}
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: ∀ {n} (v : fixed_list A n), (∀ (l : list A) {n : nat} (h : length l = n), P (tag l h)) → P v
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| n (tag l h) H := @H l n h
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definition nil : fixed_list A 0 :=
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tag [] rfl
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lemma length_succ {n : nat} {l : list A} (a : A) : length l = n → length (a::l) = succ n :=
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λ h, congr_arg succ h
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definition cons {n : nat} : A → fixed_list A n → fixed_list A (succ n)
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| a (tag v h) := tag (a::v) (length_succ a h)
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notation a :: b := cons a b
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protected definition is_inhabited [instance] [h : inhabited A] : ∀ (n : nat), inhabited (fixed_list A n)
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| 0 := inhabited.mk nil
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| (succ n) := inhabited.mk (inhabited.value h :: inhabited.value (is_inhabited n))
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protected definition has_decidable_eq [instance] [h : decidable_eq A] : ∀ (n : nat), decidable_eq (fixed_list A n) :=
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_
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definition head {n : nat} : fixed_list A (succ n) → A
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| (tag [] h) := by contradiction
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| (tag (a::v) h) := a
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definition tail {n : nat} : fixed_list A (succ n) → fixed_list A n
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| (tag [] h) := by contradiction
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| (tag (a::v) h) := tag v (succ.inj h)
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theorem head_cons {n : nat} (a : A) (v : fixed_list A n) : head (a :: v) = a :=
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by induction v; reflexivity
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theorem tail_cons {n : nat} (a : A) (v : fixed_list A n) : tail (a :: v) = v :=
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by induction v; reflexivity
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theorem head_lcons {n : nat} (a : A) (l : list A) (h : length (a::l) = succ n) : head (tag (a::l) h) = a :=
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rfl
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theorem tail_lcons {n : nat} (a : A) (l : list A) (h : length (a::l) = succ n) : tail (tag (a::l) h) = tag l (succ.inj h) :=
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rfl
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definition last {n : nat} : fixed_list A (succ n) → A
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| (tag l h) := list.last l (ne_nil_of_length_eq_succ h)
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theorem eta : ∀ {n : nat} (v : fixed_list A (succ n)), head v :: tail v = v
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| 0 (tag [] h) := by contradiction
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| 0 (tag (a::l) h) := rfl
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| (n+1) (tag [] h) := by contradiction
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| (n+1) (tag (a::l) h) := rfl
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definition of_list (l : list A) : fixed_list A (list.length l) :=
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tag l rfl
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definition to_list {n : nat} : fixed_list A n → list A
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| (tag l h) := l
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theorem to_list_of_list (l : list A) : to_list (of_list l) = l :=
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rfl
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theorem to_list_nil : to_list nil = ([] : list A) :=
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rfl
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theorem length_to_list {n : nat} : ∀ (v : fixed_list A n), list.length (to_list v) = n
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| (tag l h) := h
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theorem heq_of_list_eq {n m} : ∀ {v₁ : fixed_list A n} {v₂ : fixed_list A m}, to_list v₁ = to_list v₂ → n = m → v₁ == v₂
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| (tag l₁ h₁) (tag l₂ h₂) e₁ e₂ := begin
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clear heq_of_list_eq,
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subst e₂, subst h₁,
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unfold to_list at e₁,
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subst l₁
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end
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theorem list_eq_of_heq {n m} {v₁ : fixed_list A n} {v₂ : fixed_list A m} : v₁ == v₂ → n = m → to_list v₁ = to_list v₂ :=
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begin
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intro h₁ h₂, revert v₁ v₂ h₁,
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subst n, intro v₁ v₂ h₁, rewrite [heq.to_eq h₁]
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end
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theorem of_list_to_list {n : nat} (v : fixed_list A n) : of_list (to_list v) == v :=
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begin
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apply heq_of_list_eq, rewrite to_list_of_list, rewrite length_to_list
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end
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/- append -/
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definition append {n m : nat} : fixed_list A n → fixed_list A m → fixed_list A (n + m)
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| (tag l₁ h₁) (tag l₂ h₂) := tag (list.append l₁ l₂) (by rewrite [length_append, h₁, h₂])
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infix ++ := append
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open eq.ops
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lemma push_eq_rec : ∀ {n m : nat} {l : list A} (h₁ : n = m) (h₂ : length l = n), h₁ ▹ (tag l h₂) = tag l (h₁ ▹ h₂)
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| n n l (eq.refl n) h₂ := rfl
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theorem append_nil_right {n : nat} (v : fixed_list A n) : v ++ nil = v :=
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induction_on v (λ l n h, by unfold [fixed_list.append, fixed_list.nil]; congruence; apply list.append_nil_right)
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theorem append_nil_left {n : nat} (v : fixed_list A n) : !zero_add ▹ (nil ++ v) = v :=
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induction_on v (λ l n h, begin unfold [fixed_list.append, fixed_list.nil], rewrite [push_eq_rec] end)
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theorem append_nil_left_heq {n : nat} (v : fixed_list A n) : nil ++ v == v :=
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heq_of_eq_rec_left !zero_add (append_nil_left v)
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theorem append.assoc {n₁ n₂ n₃} : ∀ (v₁ : fixed_list A n₁) (v₂ : fixed_list A n₂) (v₃ : fixed_list A n₃), !add.assoc ▹ ((v₁ ++ v₂) ++ v₃) = v₁ ++ (v₂ ++ v₃)
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| (tag l₁ h₁) (tag l₂ h₂) (tag l₃ h₃) := begin
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unfold fixed_list.append, rewrite push_eq_rec,
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congruence,
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apply list.append.assoc
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end
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theorem append.assoc_heq {n₁ n₂ n₃} (v₁ : fixed_list A n₁) (v₂ : fixed_list A n₂) (v₃ : fixed_list A n₃) : (v₁ ++ v₂) ++ v₃ == v₁ ++ (v₂ ++ v₃) :=
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heq_of_eq_rec_left !add.assoc (append.assoc v₁ v₂ v₃)
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/- reverse -/
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definition reverse {n : nat} : fixed_list A n → fixed_list A n
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| (tag l h) := tag (list.reverse l) (by rewrite [length_reverse, h])
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theorem reverse_reverse {n : nat} (v : fixed_list A n) : reverse (reverse v) = v :=
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induction_on v (λ l n h, begin unfold reverse, congruence, apply list.reverse_reverse end)
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theorem fixed_list0_eq_nil : ∀ (v : fixed_list A 0), v = nil
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| (tag [] h) := rfl
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| (tag (a::l) h) := by contradiction
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/- mem -/
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definition mem {n : nat} (a : A) (v : fixed_list A n) : Prop :=
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a ∈ elt_of v
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notation e ∈ s := mem e s
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notation e ∉ s := ¬ e ∈ s
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theorem not_mem_nil (a : A) : a ∉ nil :=
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list.not_mem_nil a
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theorem mem_cons [simp] {n : nat} (a : A) (v : fixed_list A n) : a ∈ a :: v :=
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induction_on v (λ l n h, !list.mem_cons)
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theorem mem_cons_of_mem {n : nat} (y : A) {x : A} {v : fixed_list A n} : x ∈ v → x ∈ y :: v :=
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induction_on v (λ l n h₁ h₂, list.mem_cons_of_mem y h₂)
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theorem eq_or_mem_of_mem_cons {n : nat} {x y : A} {v : fixed_list A n} : x ∈ y::v → x = y ∨ x ∈ v :=
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induction_on v (λ l n h₁ h₂, eq_or_mem_of_mem_cons h₂)
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theorem mem_singleton {n : nat} {x a : A} : x ∈ (a::nil : fixed_list A 1) → x = a :=
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assume h, list.mem_singleton h
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/- map -/
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definition map {n : nat} (f : A → B) : fixed_list A n → fixed_list B n
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| (tag l h) := tag (list.map f l) (by clear map; substvars; rewrite length_map)
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theorem map_nil (f : A → B) : map f nil = nil :=
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rfl
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theorem map_cons {n : nat} (f : A → B) (a : A) (v : fixed_list A n) : map f (a::v) = f a :: map f v :=
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by induction v; reflexivity
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theorem map_tag {n : nat} (f : A → B) (l : list A) (h : length l = n)
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: map f (tag l h) = tag (list.map f l) (by substvars; rewrite length_map) :=
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by reflexivity
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theorem map_map {n : nat} (g : B → C) (f : A → B) (v : fixed_list A n) : map g (map f v) = map (g ∘ f) v :=
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begin cases v, rewrite *map_tag, apply subtype.eq, apply list.map_map end
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theorem map_id {n : nat} (v : fixed_list A n) : map id v = v :=
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begin induction v, unfold map, congruence, apply list.map_id end
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theorem mem_map {n : nat} {a : A} {v : fixed_list A n} (f : A → B) : a ∈ v → f a ∈ map f v :=
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begin induction v, unfold map, apply list.mem_map end
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theorem exists_of_mem_map {n : nat} {f : A → B} {b : B} {v : fixed_list A n} : b ∈ map f v → ∃a, a ∈ v ∧ f a = b :=
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begin induction v, unfold map, apply list.exists_of_mem_map end
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theorem eq_of_map_const {n : nat} {b₁ b₂ : B} {v : fixed_list A n} : b₁ ∈ map (const A b₂) v → b₁ = b₂ :=
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begin induction v, unfold map, apply list.eq_of_map_const end
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/- product -/
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definition product {n m : nat} : fixed_list A n → fixed_list B m → fixed_list (A × B) (n * m)
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| (tag l₁ h₁) (tag l₂ h₂) := tag (list.product l₁ l₂) (by rewrite [length_product, h₁, h₂])
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theorem nil_product {m : nat} (v : fixed_list B m) : !zero_mul ▹ product (@nil A) v = nil :=
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begin induction v, unfold [nil, product], rewrite push_eq_rec end
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theorem nil_product_heq {m : nat} (v : fixed_list B m) : product (@nil A) v == (@nil (A × B)) :=
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heq_of_eq_rec_left _ (nil_product v)
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theorem product_nil {n : nat} (v : fixed_list A n) : product v (@nil B) = nil :=
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begin induction v, unfold [nil, product], congruence, apply list.product_nil end
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theorem mem_product {n m : nat} {a : A} {b : B} {v₁ : fixed_list A n} {v₂ : fixed_list B m} : a ∈ v₁ → b ∈ v₂ → (a, b) ∈ product v₁ v₂ :=
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begin cases v₁, cases v₂, unfold product, apply list.mem_product end
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theorem mem_of_mem_product_left {n m : nat} {a : A} {b : B} {v₁ : fixed_list A n} {v₂ : fixed_list B m} : (a, b) ∈ product v₁ v₂ → a ∈ v₁ :=
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begin cases v₁, cases v₂, unfold product, apply list.mem_of_mem_product_left end
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theorem mem_of_mem_product_right {n m : nat} {a : A} {b : B} {v₁ : fixed_list A n} {v₂ : fixed_list B m} : (a, b) ∈ product v₁ v₂ → b ∈ v₂ :=
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begin cases v₁, cases v₂, unfold product, apply list.mem_of_mem_product_right end
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/- ith -/
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open fin
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definition ith {n : nat} : fixed_list A n → fin n → A
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| (tag l h₁) (mk i h₂) := list.ith l i (by rewrite h₁; exact h₂)
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lemma ith_zero {n : nat} (a : A) (v : fixed_list A n) (h : 0 < succ n) : ith (a::v) (mk 0 h) = a :=
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by induction v; reflexivity
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lemma ith_fin_zero {n : nat} (a : A) (v : fixed_list A n) : ith (a::v) (zero n) = a :=
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by unfold zero; apply ith_zero
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lemma ith_succ {n : nat} (a : A) (v : fixed_list A n) (i : nat) (h : succ i < succ n)
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: ith (a::v) (mk (succ i) h) = ith v (mk_pred i h) :=
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by induction v; reflexivity
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lemma ith_fin_succ {n : nat} (a : A) (v : fixed_list A n) (i : fin n)
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: ith (a::v) (succ i) = ith v i :=
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begin cases i, unfold fin.succ, rewrite ith_succ end
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lemma ith_zero_eq_head {n : nat} (v : fixed_list A (nat.succ n)) : ith v (zero n) = head v :=
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by rewrite [-eta v, ith_fin_zero, head_cons]
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lemma ith_succ_eq_ith_tail {n : nat} (v : fixed_list A (nat.succ n)) (i : fin n) : ith v (succ i) = ith (tail v) i :=
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by rewrite [-eta v, ith_fin_succ, tail_cons]
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protected lemma ext {n : nat} (v₁ v₂ : fixed_list A n) (h : ∀ i : fin n, ith v₁ i = ith v₂ i) : v₁ = v₂ :=
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begin
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induction n with n ih,
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rewrite [fixed_list0_eq_nil v₁, fixed_list0_eq_nil v₂],
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rewrite [-eta v₁, -eta v₂], congruence,
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show head v₁ = head v₂, by rewrite [-ith_zero_eq_head, -ith_zero_eq_head]; apply h,
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have ∀ i : fin n, ith (tail v₁) i = ith (tail v₂) i, from
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take i, by rewrite [-ith_succ_eq_ith_tail, -ith_succ_eq_ith_tail]; apply h,
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show tail v₁ = tail v₂, from ih _ _ this
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end
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/- tabulate -/
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definition tabulate : Π {n : nat}, (fin n → A) → fixed_list A n
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| 0 f := nil
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| (n+1) f := f (@zero n) :: tabulate (λ i : fin n, f (succ i))
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theorem ith_tabulate {n : nat} (f : fin n → A) (i : fin n) : ith (tabulate f) i = f i :=
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begin
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induction n with n ih,
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apply elim0 i,
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cases i with v hlt, cases v,
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{unfold tabulate, rewrite ith_zero},
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{unfold tabulate, rewrite [ith_succ, ih]}
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end
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end fixed_list
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